Average Error: 14.7 → 0.0
Time: 1.2m
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2077704617075050.2:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 3907.4508450000635:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2077704617075050.2:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 3907.4508450000635:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\

\end{array}
double f(double x) {
        double r20065896 = x;
        double r20065897 = r20065896 * r20065896;
        double r20065898 = 1.0;
        double r20065899 = r20065897 + r20065898;
        double r20065900 = r20065896 / r20065899;
        return r20065900;
}
↓
double f(double x) {
        double r20065901 = x;
        double r20065902 = -2077704617075050.2;
        bool r20065903 = r20065901 <= r20065902;
        double r20065904 = 1.0;
        double r20065905 = 5.0;
        double r20065906 = pow(r20065901, r20065905);
        double r20065907 = r20065904 / r20065906;
        double r20065908 = r20065904 / r20065901;
        double r20065909 = r20065907 + r20065908;
        double r20065910 = r20065901 * r20065901;
        double r20065911 = r20065910 * r20065901;
        double r20065912 = r20065904 / r20065911;
        double r20065913 = r20065909 - r20065912;
        double r20065914 = 3907.4508450000635;
        bool r20065915 = r20065901 <= r20065914;
        double r20065916 = fma(r20065901, r20065901, r20065904);
        double r20065917 = r20065901 / r20065916;
        double r20065918 = r20065915 ? r20065917 : r20065913;
        double r20065919 = r20065903 ? r20065913 : r20065918;
        return r20065919;
}

Error

Bits error versus x

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2077704617075050.2 or 3907.4508450000635 < x

    1. Initial program 30.4

      \[\frac{x}{x \cdot x + 1}\]

    2. Simplified30.4

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]

    3. Taylor expanded around -inf 0.0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]

    4. Simplified0.0

      \[\leadsto \color{blue}{\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}}\]

    if -2077704617075050.2 < x < 3907.4508450000635

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2077704617075050.2:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 3907.4508450000635:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))